Question: $ 3.\overline{1} \div 0.\overline{7} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= 31.1111...\\ x &= 3.1111...\end{align*} $ $\begin{align*} 9x &= 28 \\ x &= \dfrac{28}{9}\end{align*} $ $\begin{align*} 10y &= 7.7777...\\ y &= 0.7777...\end{align*} $ $\begin{align*} 9y &= 7 \\ y &= \dfrac{7}{9}\end{align*} $ So, the problem becomes: $ \dfrac{28}{9} \div \dfrac{7}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{28}{9} \times \dfrac{9}{7} = {?} $ $ \phantom{\dfrac{28}{9} \times \dfrac{7}{9}} = \dfrac{28 \times 9}{9 \times 7} $ $ \phantom{\dfrac{28}{9} \times \dfrac{7}{9}} = \dfrac{28 \times \cancel{9}} {\cancel{9} \times 7} $ $ \phantom{\dfrac{28}{9} \times \dfrac{7}{9}} = \dfrac{28}{7} $ Simplify: ${= 4}$